Received 5 December 2015; accepted 24 April 2016; published 27 April 2016
We assume that on the interval, , a < b. The following mesh
is given, where N is a natural number. Let Pk be the set of polynomials of degree ≤ k and Сk[a, b] be the set of continuous on the [a, b] functions having continuous derivative of order k,. In the book of Stechkin and Subbotin  the following is given.
Definition. The function is called by interpolation cubic spline with respect to the mesh (1) for the function F(x), if:
The points are called by the nodes of the spline.
Later on for convenience we let and the obtained results will remain valid for any finite interval [a, b].
Let be independent identical distributed random variables with unknown density distribution f(x) concentrated and continuous on the interval [0, 1], and SN(x) be cubic spline interpolating the values yk = Fn(xk) in the points xk = kh, , N=N(n) with “boundary conditions”
Here Fn(x) is the empirical function of the distribution of the sample, and,
as, and are given real numbers. Concrete choice of these numbers depends on the considered problem.
As estimation of an unknown probability density we take the statistics.
In the present work as estimation of the unknown density f(x) we take the statistics defined as in Theorem 1 and in Theorem 2 as well.
It is clear that, in Theorems 1 and 2 spline estimations are constructed with different boundary conditions.
Theorem 3 is devoted to asymptotic unbiasedness of the spline estimation. Also for completeness of the results the dispersion and the covariance of the spline-estimation are given.
In the main Theorem 4 necessity and sufficiency conditions for strong consistency of the spline-estimation are given.
Similar result for the Persen-Rozenblatt estimation is obtained in the book of Nadaraya (1983)  .
More detailed review on spline estimation is given in works of Wegman, Wright  , Muminov  .
2. Auxiliary Results
Using the results of the work Lii  the following theorems are easily proved.
2.1. Theorem 1
Let Fn(x) be empirical function of the distribution constructed by simple sample and SN(x) be cubic spline interpolating the values Fn(xk) in the nodes of the mesh (1). If we choose the boundary conditions for SN(x) in the form
then the derivative of the spline function is defined by the equality
Here, for, 0
Ci,j(x) are defined by the following relations:
, for the other i and j.
2.2. Theorem 2
Let Fn(x) be empirical function of the distribution constructed by simple sample and SN(x) be cubic spline interpolating the values Fn(xk). in the mesh (1). If we choose the boundary conditions for SN(x) in the form
Then the derivative of the spline function is defined by the equality
where, for, ,
and Ci,j are defined by formula (2).
We introduce the following denotations:
is the simple sample from the general population
is empirical function of distribution of the sample;
is the empirical process;
is the sequence of wiener processes;
is the brownian bridge.
We give the auxiliary lemmas.
2.3. Lemma 1 
There exists a probability space (Ω, F, P).
On which it can be defined version and the sequence of Brownian bridges Bn(t) such that for all x > 0
where a = 3.26, b = 4.86, с = 2.70.
2.4. Lemma 2 
Let be modulus of continuity of the brownian bridge Bn(t),
and. Then with probability 1 does not exceed the quantity.
Here is the random variable which is not less than 1 almost everywhere and.
3. Main Results and Proofs
The following theorem characterizes the asymptotic behavior of the bias, the covariance and the dispersion of the spline estimation.
3.1. Theorem 3
Let be the spline estimation.
1) If and are defined as in Theorem 2, then for
2) If and are defined as in Theorem 1, then
where 0 < x < 1,
[y] is the integer part of the number y.
3) Suppose, , , d = i ? j, and, then for
Proof. By virtue of, Theorems 9, 11, 12 from Stechkin and Subbotin  and Theorems 1 from Lii  follows the first statement of Theorem 3. The second and the third statement of Theorem 3 are proved in Lii  .
3.2. Theorem 4
Suppose as. Then in order with probability 1
it is necessary and sufficient that the function g(x) is the density of the distribution F(x) concentrated and continuous on the interval [0,1] with respect to Lebesgue measure.
Proof. Sufficiency. It is clear that
First we estimate the term in the right hand part of (3). We have
From Lemma 1 it follows that with probability 1 for
If we denote the modulus of continuity by then from
with probability and
This, combining (3)-(6) and using Theorem 3 we get the sufficiency condition of Theorem 4.
Necessity. Let with probability 1
Hence, from continuity of it follows continuity of g(x) on the interval [0, 1].
Therefore, the sequence random variables
are uniformly integrable. Therefore according to Theorem 5 from Shiryaev  and the inequalities
it follows that for
By virtue of (7) it is easy to see that the sequence of functions
uniformly converges to some continuous function g0(x), i.e. for
We show now continuity of F(x) on the interval [0, 1].
We assume the inverse that there exists a point x0, such that. Then by virtue of (8) and
it follows continuity of F(x) on the interval [0, 1].
By (8) for all
From another side, according to Theorem 11 from Stechkin and Subbotin (1976)
By virtue of (9)-(11)
Theorem 4 is proved.